eugene@eos.UUCP (Eugene Miya) (02/16/90)
This is a good question.
To the fellow at CMU who asked, the reference previously posted was
%A A. Michael Noll
%T A Computer Technique for Displaying n-Dimensional Hyperobjects
%J CACM
%V 10
%N 8
%D August 1967
%P 469-473
Side note:
1) if you the current reader, don't know CACM: spend some time learning.
Do some non-graphics computing.
2) if you don't know about refer format above, spend some time learn there
as well. [Databases and inverted indices.] If you aren't on a Unix
system, 8).
This is a reasonable reference to the problem. The net provided this
reference, so good people do read. Figure 3 is a tesseract rotation
(wireframe). Page 472's last paragraph starts: A five-dimensional
hypercube was projected, ..However, the final...projection was..
extremely complicated.
Next page, discussion:
At first it was thought ..result in .. some "feeling" or insight...
Unfortunately, this did not happen, and we were still as puzzled as the
inhabitants of Flatland...
Goes on to say the value is in movies (time varying) and I know that. Etc. etc.
Sure useful to that degree.
So what beyond Flatland by Abbott? Let me suggest a couple of things.
I will also ask my climbing and skiing partner Burke.
I mentioned Wolfe. This book isn't easy, it's not even mostly 4D but it will
make you think:
%A Harold Wolfe
%T Non-Euclidean Geometry
%I Holt, Rinehart and Winston
%D 1945
You have to KNOW your Euclidean geometry.
I suggest the above reference and book for the 1 or 2 who take the time to
do such things. I know most netters to be too lazy to get away from
keyboards. You have to begin to think that such geometries exist.
Unfortunately, I remember a great diagram, an ellipse cut with two "parallel"
lines in the Euclidean sense. Then several other lines intersecting
at one point on one of the lines: that we may exist in a Non-Euclidean
world. Page 108, figure 55 is something like this. The "lazies" have
no need to read this, since they will not know as those who see this
diagram. Page 201 is the start of the chapter on the Consistency of
Non-Euclidean Geometries. You will have to work at this book.
You have to know your trig well.
Another thing to consider are some of the writings and math Escher did
to do some of his drawings. On my wall at home is an ancient instrument:
a T-square, from my per-computer days. There were at least 3 non-standard
drawing boards which used special perspective T-squares. These are interesting
learning tools. Using them can help to constitute an interesting learning
experience. This might be an interesting analog way of doing some of this
(remember paper is still 2-D).
Escher is a real inspiration to some. Burke, who is currently
teaching a class at UCSC was really inspired by Escher (decades pre-GEB)
recently bemoaned the "fact" that there are no hackers in his classes
anymore (people more worried about getting jobs, grades, etc. perhaps).
He wonders if hacking (not that he ever was a hacker himself 8)
[learning for learning's sake] has died. But then he said that with
respect to his quaterion calculator. 8)
Try Wolfe, I'll ask Bill for his refs.
Another gross generalization from
--eugene miya, NASA Ames Research Center, eugene@aurora.arc.nasa.gov
resident cynic at the Rock of Ages Home for Retired Hackers:
"You trust the `reply' command with all those different mailers out there?"
"If my mail does not reach you, please accept my apology."
{ncar,decwrl,hplabs,uunet}!ames!eugene
Do you expect anything BUT generalizations on the net?
[If it ain't source, it ain't software -- D. Tweten]ruffwork@mist.cs.orst.edu (Ritchey Ruff) (02/17/90)
two good references, eugene. another good one is
Hypergraphics: Visualizing Complex Relationships
in Art, Science and Technology, Edited by David W. Brisson,
AAAI Selected Symposium 24, Westview Press:Boulder Co.,
ISBN: 0-89158-292-4.
this book has lots of interesting n-dimensional stuff.
from "Complex Relations in Urban and Regional Planning: An
Application of Hypergraphics" to "An Impossible 4 Dimensional
Illusion."
--ritchey ruff ruffwork@cs.orst.edu
...there is something extremely appealing about the
concept of Barry Manilow at 45 degrees below zero.
--- Dave Barry ("It's Tax Time", 2-12-90)